A liapounov function for an automomous second-order ordinary differential equation

Abstract

Two results on the stability of a singular point of the differential equation

$$x'' = f(x)(1 + x'^2 ) + g(x)\surd (1 + x'^2 ) + h(x,x')$$

(1)

are proved, in the form of conditions on the functions f, g, h. The results are in the style of many results for Lienard's equation, and the proofs rest on the choice of a suitable Liapounov function. The arguments leading to this choice may be extended to any equation "close" to one for which a first integral may be obtained.